Delone Sets and Polyhedral Tilings: Local Rules and Global Order

Summary form only given. An appropriate concept for describing an arbitrary discrete atomic structure is the Delone set (or an (r,R)-system). Structures with long-range order such as crystals involves a concept of the space group as well. A mathematical model of an ideal monocrystalline matter is defined now as a Delone set which is invariant with respect to some space group. One should emphasize that under this definition the wellknown periodicity of crystal in all 3 dimensions is not an additional requirement. By the celebrated Schoenflies-Bieberbach theorem, any space group contains a translational subgroup with a finite index. Thus, a mathematical model of an ideal crystal uses two concepts: a Delone set (which is of local character) and a space group (which is of global character). Since the crystallization is a process which results from mutual interaction of just nearby atoms, it is believed (L. Pauling, R. Feynmann et al) that the long-range order of atomic structures of crystals (and quasi-crystals too) comes out local rules restricting the arrangement of nearby atoms. However, before 1970s there were no whatever rigorous results until Delone and his students (Dolbilin, Stogrin, Galiulin) initiated developing the local theory of crystals. The main aim of this theory was (and is) rigorous derivation of space group symmetry of a crystalline structure from the pair-wise identity of local arrangements around each atoms. To some extent, it is analogous to that as, in due time, it was rigorously proved that space group symmetry contains a translational subgroup. In the talk it is supposed to expose some results on local rules for crystals obtained by Delone, Dolbilin, Stogrin, and their followers and to outline the frontier between crystalline and quasi-crystalline local rules.